The present invention relates to an apparatus for identifying a mathematical relation between variables of measured data. An apparatus for identifying a mathematical relation identifies a mathematical relation between numerical data of, for example, a form, trajectory, etc. The extrapolation, interpolation, storing of such data can be facilitated
For example, the apparatus can be used as follows:
1) Expression of a shape of an object
In a designing of a shape of an automobile, data of a shape of an object can be stored as a set of numerical data. However, when a shape can be expressed as a mathematical expression including a plurality of parameters, data can be easily extrapolated and interpolated. And new shape can be easily obtained, by varying the value of some parameters. And a computer analysis of the new shape, for example, aerodynamical resistance, etc, becomes easy. Further, a preparation of a mold for fabrication of an object having the corresponding shape, using a numerically controlled working machine, becomes easy.
In a shoe shop, even when a client found a pair of shoes, the size of which is identical to the size of his feet, and the design of which is acceptable, if the pair of shoes do not fit actually to his feet, he will not buy the pair of shoes. There are many variety in the shape of foot of human being. Therefore, sizes including small number of measured values of a foot are not sufficient to express a shape of a foot of human being. No apparatus or method is known, which can determine values of parameters of a mathematical expression, representing a complex shape of an object, for example, a shape of shoe or a shape of foot of human being.
2) Identification of a mathematical relation between variables of experimental data, when the mathematical relation shall be expressed by a non-linear function:
Even when a mathematical relation between obtained data is non-linear, data are often extrapolated and interpolated, using a linear approximation. However, when a mathematical relation between the variables of data can be identified, data can be exactly extrapolated and interpolated, without using linear approximation.
Assuming that a food is produced from stuffs A, B. And a characteristic value of the food X, for example, concentration Z of an amino acid, is a function of the concentration X, Y of the stuffs A, B, the pressure P and temperature T of a treatment. When the mathematical relation between these variables can be identified, it becomes easy to obtain the most preferable values of the X, Y. P, T, for getting the most appropriate concentration Z, using the mathematical relation. Thus the development of a new product can be facilitated.
For example, assuming that the relation between the variables X, Y, P, T, Z is as follow, no apparatus and method is known, which can determine the coefficients K0, K1, K2, K3, by small times of experiments.
Z/(Xxc2x7Y)=K0+K1xc2x7P1xc2x7K2T+K3xc2x7X
3) Representation of a trajectory of a moving object: For example, it is said that the trajectory of golf club head of a professional golfer is approximately a plane, on the other hand, the trajectory of golf club head of an ordinal amateur is not a plane. The trajectory of golf club head is not a line, thus the mathematical expression of the trajectory of golf club head is not simple. An apparatus for identifying the mathematical relation identifies the mathematical relation of a trajectory of such a moving object so as to analyze the trajectory.
Trajectories of an airplane flying around over an airport, a ship sailing on the ocean, or a car running on the road, are not linear, but the changing rate of the direction of the movement is slow. When the mathematical relation of the motion of such moving objects can be identified, it is possible to estimate the possibility of a collision of such moving objects.
Some methods for identifying the mathematical relation between measured values are proposed. For example, a least square root method for determining the factors a and b in a linear relation y=ax+b is widely employed, under the assumption that a linear relation y=ax+b stands between the input data. Also a least square root method after a logarithmic transformation is widely employed.
Methods for identifying the mathematical relation between the measured data are explained compactly in xe2x80x9cStatistics for analytical chemistryxe2x80x9d by J. C. Miller/J. N. Miller, which is translated into Japanese by Munemori Makoto and published in Japan from Kyouritu Shuppan in 1991.
Japanese patent application JP-5-334431-A discloses an apparatus for giving a mathematical function, which approximates data of points on a line This apparatus is not applicable, when the data is not data of points on a line, or when the data is data of over three dimension.
Japanese patent application JP-5-266063-A discloses an apparatus for interpolating data in n dimension space, using a super surface in a (n+1) dimension space. This apparatus can interpolate data, but does not identify the mathematical relation of the data.
An object of the present invention is to propose an apparatus and method for outputting a mathematical relation between base variables (x1, x2, . . . , xp), when a set of input data d is comprised of p base variables (x1, x2, . . . , xp), and a plurality of data sets d(i) of such a data set d are inputted, where the symbols xe2x80x9cixe2x80x9d is a parameter for distinguishing the sets of data set.
The meaning of symbols and terms used in this Specification and the Claims are explained, before explaining the present invention. Symbol xe2x80x9c{circumflex over ( )}xe2x80x9d is a power (far example, (xe2x88x922){circumflex over ( )}3=xe2x88x928). Power is expressed also using a suffix (for example, (xe2x88x922)3=xe2x88x928).
Symbol xe2x80x9c.xe2x80x9d is a multiplication (for example 2xc2x73=6). However, this symbol is abbreviated, when the meaning is obvious (for example, 2x+3y=0 means 2xc2x7x+3xc2x7y=0).
Symbol xe2x80x9c less than axc2x7b greater than xe2x80x9d is an inner product (scalar product) of vectors.
xe2x80x9cBase variables x1, . . . , xpxe2x80x9d are names of variables of data inputted into the apparatus according to the present invention (for example coordinates x, y, z pressure p, or time t).
xe2x80x9cInput data dxe2x80x9d is a set of base variables (x1, x2, . . . xp).
xe2x80x9cd=(x1, . . . , xp)xe2x80x9d means that the number of the base variables of the input data d is p, and the final base variable is xp.
xe2x80x9cInput data d(i)xe2x80x9d is the i-th set of the input data d. Symbol xe2x80x9cixe2x80x9d is called xe2x80x9cdata specifying parameterxe2x80x9d.
xe2x80x9cxjixe2x80x9d is the value of the j-th variable of the i-th set of data d(i).
xe2x80x9cMathematical function programxe2x80x9d is a program for outputting a value corresponding to input reference values, after executing a mathematical calculation or using a table.
xe2x80x9cAn input reference and a function specifying referencexe2x80x9d: For example, a function exp(kxc2x7x) can be considered as a two variable function, in such a case, the function exp(kxc2x7x) has two input variables k and x. However, this function can be considered also as a one variable function exp(kxc2x7x). In this case, the function has an input reference x and a reference k for specifying the function. References for specifying the function are called xe2x80x9cfunction specifying referencexe2x80x9d, and references as input values are called xe2x80x9cinput referencexe2x80x9d.
xe2x80x9cFunction specifying parameter mxe2x80x9d is a parameter for specifying a mathematical function program gm.
xe2x80x9cBase functionxe2x80x9d is a function gk in a set of mathematical function programs gm stored in a mathematical function program storing memory, and specified by the function specifying parameter (m=k). The input variables to be inputted to the input reference of the mathematical function gk are specified. The function specifying references are specified when it is necessary. When the function is a constant function, which outputs a constant value irrespectively to the input, it is not necessary to specify the input references.
xe2x80x9cCandidate mathematical relationxe2x80x9d is a set of base functions (f1, . . . , fq).
In the present invention, a mathematical relation is identified as a linear combination of base functions.
xe2x80x9cA temporal candidate mathematical relationxe2x80x9d is a set of base functions selected temporarily as a candidate mathematical relation, when the most appropriate mathematical relation is sought, by changing the candidate mathematical relation.
xe2x80x9c(f1, . . . , fq)xe2x80x9d means that the candidate mathematical relation is comprised of q base functions, and the final base function is fq.
xe2x80x9cVector component Fkixe2x80x9d is a value of a base function fk, corresponding to the values of the base variable (x1, . . . , xp) of the i-th input data d(i) inputted to the input reference of the base function fk.
xe2x80x9cVector F(i)xe2x80x9d is a vector in a q dimensional space comprised of vector components Fki (k=1, . . . , q).
xe2x80x9cVector component array VE_ARRAYxe2x80x9d is a two dimensional array storing vector components Fki.
xe2x80x9cVector space Fxe2x80x9d is a space spanned by q dimensional vectors F(i).
xe2x80x9cPlanexe2x80x9d is a set of points in a q dimensional vector space, which satisfies the following relation, where the L1, L2, . . . , Lq are constant (Appendix B):
L1xc2x7x1+ . . . +Lqxc2x7xq=0 xe2x80x83xe2x80x83(1)
xe2x80x9cDimension cosinexe2x80x9d is a set of numerals (L1, . . . , Lq) which satisfy the expression (1), and normalized as follows:
(L1)2+(L2)2++(Lq)2=1xe2x80x83xe2x80x83(2)
xe2x80x9cMapping planexe2x80x9d is a mapping of a set of input data d(i) into the vector space F, and the mapping is a plane or nearly a plane. Also, a plane the square sum of the perpendicular lines from mapped points to the plane is zero or nearly equal to zero (namely, smaller than a predetermined valuexcex0), is called a mapping plane, (Appendix A).
xe2x80x9cPlane degree index PIxe2x80x9d is an index indicating in what degree the vectors F(1), . . . , F(q) can be regarded as a plane. Plane degree index can be defined on any industrial or commercial ground (Appendix D).
xe2x80x9cCorrelation sum  less than a|b greater than xe2x80x9d is a sum of products of a vector component Fai obtained from a base function fa and a vector component Fbi obtained from a base function fb, the sum is carried out in respect with the input data specifying parameter i (Appendix F).
 less than a|b greater than =xcexa3(Faixc2x7Fbi)xe2x80x83xe2x80x83(3a)
The term  less than a|b greater than  means also the sum of the products multiplied by a square of weight Wi2:
 less than a|b greater than =xcexa3Wi2(Faixc2x7Fbi)xe2x80x83xe2x80x83(3b)
xe2x80x9cCorrelation sum matrix Cxe2x80x9d is a matrix, the elements of which are correlation sums.
The object of the present invention is attained by the apparatus for identifying the mathematical relation according to claim 1, and the method for identifying the mathematical relation according to claim 10. The principle of the present invention is explained in Appendix A.
A series of data d(i) comprised of p basic variables xj (j=1, . . . , p) are acquired by a data acquisition means. The unit of data is d(i)=(x1i, x2i, . . . xpi).
A plurality of mathematical function programs gm are stored in a mathematical program storing memory
A base function defining means defines a base function by specifying a specific mathematical function program gk in the mathematical function program gm in the mathematical program storing memory, by specifying the parameter (m=k), and further specifying the base variables to be inputted into the input reference of the mathematical function Also mathematical function specifying parameters are specified, when it is necessary.
A candidate mathematical relation specifying means specifies a temporal mathematical relation by specifying a set of base functions (f1, . . . , fq). In this invention, the mathematical relation is identified as a combination of those base functions (f1, . . . , fq) It is possible to design the candidate mathematical relation specifying means to specify a series of temporal candidate mathematical relations sequentially.
A vector component acquisition means sends the values of the basic variables to each of the input references of base functions fk (k=1, . . . , q) included in the candidate mathematical relation, for each data set d(i). The vector (component acquisition means gets the output value Fki of the base functions fk and stores them into a two dimensional array. The array (is called VC_ARRAY. When the base functions are fk (k=1, . . . , q), and s sets of data are sent, the vector component array VC_ARRAY is a two dimensional array of (qxc3x97s) size.
A set of Fji is considered as a vector F(i). And the values of the functions are stored in the vector component array VC_ARRAY for every vector F(i), as follows:
F(1)=(F11,F21, . . . , Fq1)
F(2)=(F12,F22, . . . , Fq2)
. . . 
When q sets of data in the array VE_ARRAY are considered as a coordinate of points P(i) in a q dimensional space F, and if the combination of the candidate mathematical relation is correct, the points shall be found in a plane (mapping plane) in the vector space F. If the candidate mathematical relation is approximately correct, those points shall distribute near to a plane (mapping plane). As explained in Appendix C, the mapping plane can be determined from a determinant of Fji, or can be determined as a plane, the square sum of perpendicular lines from the points P(i) to the plane is the minimum.
A direction cosine acquisition means calculates and outputs the direction cosine (L1, . . . , Lq) of a plane on which the points P(i) (i=1, . . . , q) are found, or a plane, the square sum of perpendicular lines from the points P(i) to the plane are the minimum.
A mathematical relation outputting means identifies the mathematical relation as a combination of the base functions fk (k=1, . . . , q), where the combination coefficients of the base functions are the direction cosine. The mathematical relation outputting means outputs the mathematical relation in a form of the following linear combination or a mathematical relation deduced from it:                                           ∑                          k              =              1                        q                    ⁢                      xe2x80x83                    ⁢                      (                          Lk              ·              fk                        )                          =        0                            (        4        )            
According to an embodiment, the direction cosine acquisition means acquires the direction cosine, using a cofactor of a determinant V of a set of values of mathematical functions Fji. The ground of this method is explained in Appendix C.
According to another embodiment, the direction cosine acquisition means acquires the direction cosine from an eigenvector of a correlation sum matrix obtained from an array of sets of the values of base functions. The ground of this method is explained in Appendix F.
In another embodiment, the base function defining means defines a plurality of candidate mathematical relations. And so far as the eigenvalue xcex calculated by the direction cosine acquisition means is not smaller than a predetermined value, another candidate mathematical relation is set as a new temporal candidate mathematical relation to repeat to try to acquire a new eigenvalue, untill the eigenvalue becomes smaller than the predetermined value. As a result, an appropriate mathematical relation can be acquired automatically.
In another embodiment, the mathematical relation identifying apparatus is incorporated in an apparatus for interpolating or extrapolating data. When an empirical formula of a system includes a plurality of parameters is required, a candidate mathematical relation of a combination of a base functions is assumed, on an empirical or theoretical ground. The empirical formula can be identified, by acquiring the direction cosine, using the apparatus according to the present invention and estimating the validity of the expression.
In another embodiment, the apparatus for identifying a mathematical relation is incorporated in an apparatus for measuring a trajectory. At first infinite sets of the coordinates of points on a trajectory of a moving object (for example, golf club head) is acquired. Previously a candidate mathematical relation is assumed on an empirical or theoretical ground. Then the direction cosine is acquired using the apparatus according to the present invention. An empirical formula of a trajectory can be obtained from the direction cosine, after estimating the validity.
In another embodiment, the apparatus for identifying a mathematical relation is incorporated in an apparatus for measuring exterior or interior form of an object, (for example, face, foot, or shoe). At first, coordinates of infinite sets of points on the surface of an object is acquired. Previously a candidate mathematical relation is assumed on an empirical or theoretical ground. Then, the direction cosine is acquired, using the apparatus according to the present invention. An empirical formula representing the form can be obtained from the direction cosine, after estimating the validity.
The present invention can be realized as a memory media, which can be read out by a computer, in which the main part of the invention according to claim 1 is coded as software.
An embodiment of the present invention is a method corresponding to the apparatus for identifying a mathematical relation according to claim 1.